Relation Between Roots and The Coefficients of a Quadratic Equation

Quadratic Polynomial

The term “quadratic polynomial” refers to a type of polynomial in which the monomial with the highest degree is of the second degree. The term “second-order polynomial” can also be used to refer to a quadratic polynomial. This indicates that the value of at least one of the variables ought to be increased to the power of two, whilst the values of the powers of the other variables ought to be less than or equal to two but greater than -1.

A quadratic polynomial can have numerous variables. However, the quadratic polynomial with only one variable, known as a univariate polynomial, is by far the most frequent type of polynomial. A parabola is the graph that is generated when a univariate quadratic polynomial is evaluated. In this post, we will learn more about quadratic polynomials, including how to discover the roots of such a polynomial, as well as look at some examples of such polynomials.

What does it mean to have a quadratic polynomial?

A polynomial is said to be quadratic if the largest power of a variable term in the expression for the polynomial is 2, as this defines a quadratic polynomial. When determining the degree of a polynomial, the only factor that is taken into account is the variable’s exponent. There is no consideration given to the influence that a coefficient or a constant term can have. A quadratic equation or a quadratic function can be derived from a quadratic polynomial when the polynomial is equated to the value 0. The answers to such equations are referred to as the roots of the quadratic equation or the zeros of the quadratic equation.

Quadratic Polynomial Definition

Quadratic Polynomial

A polynomial of the second degree that is considered to be quadratic is one in which the value of the term with the highest degree is equal to 2. In its most common representation, a quadratic equation takes the form ax² + bx + c = 0. Here, a and b are considered to be coefficients, x is the variable that is not known, and c is the term that is constant. Due to the presence of a quadratic polynomial in this equation, resolving it will result in the production of two solutions. This suggests that there are two possible values for the variable x.

Quadratic Polynomial Example

Suppose we have a quadratic polynomial x² + 4x + 4 = 0. Then, in order to determine the answers to these equations, we need to factor them as follows: (x + 2)(x + 2) = 0. As a consequence, the solutions to this quadratic equation are going to be x = -2 and -2.

The discriminant is denoted by the number b² – 4ac. D stands for it. The discriminant can be used to ascertain the roots’ nature.

Relation Between Roots And The Coefficients Of A Quadratic Equation

Take the quadratic equation ax² + bx + c = 0 where a ( ≠ 0 ) is the coefficient of x², b the coefficient of x, and c is the constant term.

Let α and β be the ax² + bx + c = 0 equation’s roots.

We will now determine how a, b, and c relate to α and β .

ax2 + bx + c now equals to 0 .

The result of multiplying both sides by 4a (a ≠ 0) i.e

4ax² + 4abx + 4ac = 0

(2ax)² + 2 * 2ax * b + b² – b² + 4ac = 0

b² – 4ac = (2ax + b)²

2ax + b = ± √b²−4ac

x = (-b ± √b²−4ac ) / 2a

Let α be equal to (-b + √b²−4ac ) / 2a

 and β = (-b – √b²−4ac ) / 2a

Therefore,

α + β = (-b + √b²−4ac ) / 2a    +   (-b – √b²−4ac ) / 2a

α + β = −2b / 2a

α + β = -b / a

sum of roots = -coefficient of x / coefficient of x² .

Once more, 

αβ = (-b + √b²−4ac ) / 2a    x   (-b – √b²−4ac ) / 2a

αβ = b²−(b²−4ac) / 4a²

αβ = 4ac / 4a²

αβ = c / a

αβ = constant term / coefficientof x² .

Therefore, the relationships between the roots (i.e., α and β ) and coefficients (i.e., a, b, and c) of the equation ax² + bx + c = 0 are represented by :

Sum of roots ( i.e., α + β) = -coefficient of x / coefficient of x² ;

Product of roots (i.e., αβ ) = constant term / coefficient of x².

Conclusion

• Sum of roots ( i.e., α + β) = -coefficient of x / coefficient of x²  
• Product of roots (i.e., αβ ) = constant term / coefficient of x²
• A polynomial with a degree of two is said to be quadratic
• Two roots are possible for a quadratic polynomial
• The formula can be used to find a quadratic equation’s roots is −b±√b²−4ac / 2a
• To determine the nature of the roots, the discriminant, equal to b2 – 4ac, is utilised
• To obtain the quadratic polynomial, insert the sum and product of the roots into the expression x² – (sum of roots)x + (product of roots)
• A quadratic polynomial has a parabola as its graph